Properties

Degree 2
Conductor $ 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s − 4-s + 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s + 10-s − 2·12-s + 13-s − 2·14-s + 2·15-s − 16-s − 5·17-s + 18-s + 6·19-s − 20-s − 4·21-s + 2·23-s − 6·24-s − 4·25-s + 26-s − 4·27-s + 2·28-s + 9·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s − 1.21·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.872·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s + 0.377·28-s + 1.67·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(121\)    =    \(11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{121} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 121,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.666156920$
$L(\frac12)$  $\approx$  $1.666156920$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.30198464662403, −18.23029015521226, −17.45129267091964, −15.94960393364486, −15.19519004143324, −14.06941374406207, −13.65039157651841, −12.93331549622801, −11.69772641079316, −9.975450561586109, −9.182457967238805, −8.356936232923193, −6.722214028918875, −5.384155723528973, −3.841779023620474, −2.752740844635733, 2.752740844635733, 3.841779023620474, 5.384155723528973, 6.722214028918875, 8.356936232923193, 9.182457967238805, 9.975450561586109, 11.69772641079316, 12.93331549622801, 13.65039157651841, 14.06941374406207, 15.19519004143324, 15.94960393364486, 17.45129267091964, 18.23029015521226, 19.30198464662403

Graph of the $Z$-function along the critical line